Rather, the problem is that at least one celebrated authority in the field hates that, and would prefer much, much more deference to authority.

I don't think this is true at all. His points against replicability are very valid and match my experience as a researcher. In particular:

Because experiments can be undermined by a vast number of practical mistakes, the likeliest explanation for any failed replication will always be that the replicator bungled something along the way.

This is a very real issue and I think that if we want to solve the current issues with science we need to be honest about this, rather than close our eyes and repeat the mantra that replication will solve everything. And it's not like he's arguing against accountability. Even in your quoted passage he says:

The field of social psychology can be improved, but not by the publication of negative findings. Experimenters should be encouraged to restrict their “degrees of freedom,” for example, by specifying designs in advance.

Now, I think he goes too far by saying that no negative findings should be published; but I think they need to be held to a high standard for the very reason he gives. On the other hand, positive findings should also be held to a higher standard.

Note that there are people much wiser than me (such as Andrew Gelman) who disagree with me; Gelman is dissatisfied with the current presumption that published research is correct. I certainly agree with this but for the same reasons that Mitchell gives, I don't think that merely publishing negative results can fix this issue.

Either way, I think you are being quite uncharitable to Mitchell.